Finite Group Databasebeta

← #41012360

Subgroups — #41012360

Order 27008.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27213745C211yesmaximal, characteristic
2#27213285C211yesmaximal, characteristic
3#27213065C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212881C211yesmaximal, characteristic
6#27212871C211yesmaximal, characteristic, complemented
7#451×C211C211yesmaximal, characteristic, complemented
8C2×#16809949—21no—
9#20269972—21no—
10#20269968—21no—
11#20269962—21no—
12#20269961—21no—
13#20269953—21no—
14#20269951C2211yescharacteristic
15(C22⋊D211)⋊C4—21no—
16C2×Dic211⋊C4—21no—
17C2×Dic211⋊C4C2211yescharacteristic
18#20269901—21no—
19C22⋊(C211⋊C8)C2211yescharacteristic
20C211⋊#95—21no—
21C211⋊#95C2211yescharacteristic
22C22×Q8×C211C2211yescharacteristic
23C2×Q16×C211—21no—
24C22⋊Q8×C211C2211yescharacteristic
25#103×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C2×D844D411yescharacteristic
28C2×C22⋊D211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C4×D211—21no—
33#16809949—42no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—42no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46C4×Dic211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—42no—
50C2×Q8×C211D411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53Q16×C211—42no—
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59D844—41no—
60C22⋊D211—41no—
61C22⋊D211—41no—
62C2×Dic211—22no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C4×D211—81no—
72C211⋊C8—41no—
73C23×C211C2×D411yescharacteristic
74Q8×C211—22no—
75Q8×C211—22no—
76Q8×C211—41no—
77Q8×C211—41no—
78Q8×C211—41no—
79C2×C4×C211—22no—
80C2×C4×C211—21no—
81C2×C4×C211—21no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C8×C211—41no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211#12011yescharacteristic
91C22×C211#13511yescharacteristic
92C22×C211#13711yescharacteristic
93C2×D211—41no—
94C2×D211—41no—
95C2×D211—41no—
96C4×C211—21no—
97C4×C211—21no—
98C4×C211—41no—
99C4×C211—42no—
100C4×C211—41no—
101Dic211—42no—
102Dic211—41no—
103Dic211—41no—
104Dic211—41no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211#45211yescharacteristic
108C2×C211#45611yescharacteristic
109C2×C211B(4,𝔽2)11yescharacteristic
110D211—81no—
111C211#160311yescharacteristic, complemented, Sylow
112#1603—2111nomaximal, Sylow
113#539—2111no—
114#451D21111yescharacteristic, complemented
115#448—2111no—
116#395—2111no—
117#328—2111no—
118#327—2111no—
119#324—2111no—
120C22×Q8C2×D21111yescharacteristic
121C2×Q16—21no—
122C2×SD16—4221no—
123#126—4221no—
124#125—4221no—
125#123—4221no—
126#123—4221no—
127C22⋊Q8—4221no—
128C22⋊Q8C2×D21111yescharacteristic
129#121—2111no—
130C4×D4—4221no—
131C2×C4⋊C4—4221no—
132C2×C4⋊C4—2111no—
133#103—21no—
134#103—4221no—
135C22⋊C8—2111no—
136C22⋊C8C2×D21111yescharacteristic
137#95—4221no—
138#95—2111no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D21111yescharacteristic
143C2×D4—4221no—
144C2×D4—2111no—
145C22×C4—21no—
146C22×C4—4221no—
147C22×C4—4221no—
148C22×C4—2111no—
149C22×C4—2111no—
150C22×C4C22×D21111yescharacteristic
151Q16—42no—
152SD16—8442no—
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—21no—
156C4⋊C4—8442no—
157C4⋊C4—4221no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4—2111no—
163C4⋊C4C22⋊D21111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—4221no—
166C22⋊C4—4221no—
167C22⋊C4—4221no—
168C22⋊C4—4221no—
169C42—4221no—
170C23—4221no—
171C23D4×D21111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—8441no—
178D4—8441no—
179D4—8441no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8441no—
189C2×C4—16881no—
190C2×C4—4222no—
191C2×C4—4221no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4D4×D21111yescharacteristic
197C2×C4C2×C22⋊D21111yescharacteristic
198C8—41no—
199C8—8441no—
200C22—21no—
201C22—21no—
202C22—8441no—
203C22—8441no—
204C22—8441no—
205C22#2026994511yescharacteristic
206C22#2027000011yescharacteristic
207C22#2027000911yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—8442no—
214C4—8441no—
215C4—8441no—
216C4—8441no—
217C2—21no—
218C2—21no—
219C2—16881no—
220C2#2721330711yescharacteristic
221C2#2721332711yescharacteristic
222C2#2721333211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#451×C211C2
socleC22×C211#120
prime radical#20269951C22
hypercenter#451D211
hypocenterC211#1603
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #27213065C2semidirect
4. #27212883C2semidirect
6. #27212871C2semidirect
7. #451×C211C2semidirect
111. C211#1603semidirect
114. #451D211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#272137452
#272132852
#272130652
#272128832
#272128812
#272128712
#451×C2112
#1603211